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Node connectivity and arc connectivity of a fuzzy graph

Sunil Mathew a,1, M.S. Sunitha b,*

a Department of Mathematics, Government Brennen College, Thalassery, Kannur 670 106, Indiab Department of Mathematics, National Institute of Technology, Calicut 673 601, India

a r t i c l e i n f o

Article history:Received 16 September 2008Received in revised form 4 September 2009Accepted 6 October 2009

Keywords:Fuzzy relationFuzzy bondsFuzzy node cutFuzzy arc cutCut-setFuzzy node connectivityFuzzy arc connectivity

a b s t r a c t

The fuzzy graph approach is more powerful in cluster analysis than the usual graph – the-oretic approach due to its ability to handle the strengths of arcs effectively. The concept ofnode-strength sequence is introduced and is studied in a complete fuzzy graph. Two newconnectivity parameters in fuzzy graphs namely, fuzzy node connectivity ðjÞ and fuzzy arcconnectivity ðj0Þ are introduced and obtained the fuzzy analogue of Whitney’s theorem.Fuzzy node cut, fuzzy arc cut and fuzzy bond are defined. Fuzzy bond is a special type ofa fuzzy bridge. It is proved that at least one of the end nodes of a fuzzy bond is a fuzzy cutn-ode. It is shown that j ¼ j0 for a fuzzy tree and it is the minimum of the strengths of itsstrong arcs. The relationships of the new parameters with already existing vertex and edgeconnectivity parameters are studied and is shown that the value of all these parameters areequal in a compete fuzzy graph. Also a new clustering technique based on fuzzy arc con-nectivity is introduced.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Fuzzy graph theory has numerous applications in various fields like clustering analysis, database theory, network anal-ysis,information theory, etc. [7]. Fuzzy models can be used in problems handling uncertainty to get more accurate and pre-cise solutions [23–25]. As in graphs, connectivity concepts play a key role in applications related with fuzzy graphs [7,19].Fuzzy graphs were introduced by Rosenfeld [9] and Yeh and Bang [21] independently in 1975. Rosenfeld in his paper ‘‘FuzzyGraphs” presented the basic structural and connectivity concepts while Yeh and Bang introduced different connectivityparameters of a fuzzy graph and discussed their applications in the paper titled ‘‘Fuzzy relations, Fuzzy graphs and their appli-cations to clustering analysis” [21].

Rosenfeld [9] has obtained the fuzzy analogues of several basic graph-theoretic concepts like bridges, paths, cycles, treesand connectedness and established some of their properties. Bhattacharya [2] has established some connectivity conceptsregarding fuzzy cutnodes and fuzzy bridges. The author has also introduced fuzzy groups and metric notion in fuzzy graphs.Bhattacharya and Suraweera [1] have introduced an algorithm to find the connectivity of a pair of nodes in a fuzzy graph [1].Also Saibal Banerjee [10] has obtained an optimal algorithm to find the degrees of connectedness in a fuzzy graph. Tong andZheng [19] have obtained an algorithm to find the connectedness matrix of a fuzzy graph. Xu [20] applied connectivityparameters of fuzzy graphs to problems in chemical structures. Fuzzy trees were characterized by Sunitha and Vijayakumar[15]. The authors have characterized fuzzy trees using its unique maximum spanning tree. A sufficient condition for a nodeto be a fuzzy cutnode is also established in [14]. Center problems in fuzzy graphs [17], blocks in fuzzy graphs [16] and prop-erties of self complementary fuzzy graphs [18] were also studied by the same authors. They have obtained a characterization

0020-0255/$ - see front matter � 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.ins.2009.10.006

* Corresponding author. Tel.: +91 0495 2286506; fax: +91 0495 2287250.E-mail addresses: sm@nitc.ac.in (S. Mathew), sunitha@nitc.ac.in (M.S. Sunitha).

1 Tel.: +91 9349769093.

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for blocks in fuzzy graphs using the concept of strongest paths [16]. Bhutani and Rosenfeld have introduced the concepts ofstrong arcs [4], fuzzy end nodes [5] and geodesics in fuzzy graphs [6]. In [4], the authors have defined the concepts of strongarcs and strong paths. They have shown the existence of a strong path between any two nodes of a fuzzy graph and havestudied the strong arcs of a fuzzy tree. In [5], the concepts of fuzzy end nodes and multimin and locamin cycles are studied.The concept of strong arc in maximum spanning trees [11] and its applications in cluster analysis and neural networks werestudied by Sameena and Sunitha [12,13,]. Sunil Mathew and Sunitha identified different types of arcs in fuzzy graphs andhave obtained an arc identification procedure [14].

In graph theory, a node in a graph G is a cut node if and only if its removal from G disconnects it. Similarly an arc is abridge if and only if its removal from G disconnects the graph G. Generalizing the concepts of cut nodes and bridges, a nodecut (vertex cut) is defined as a set of nodes whose removal from the graph disconnects it and an arc cut (cut set) is defined asa set of arcs whose removal from G disconnects G. Node connectivity of a graph is the number of nodes in a minimal (havingminimum number of nodes) node cut and arc connectivity, the number of arcs in a minimal arc cut. Fuzzy graph is a gen-eralization of classical graph. One of the notable difference between graphs and fuzzy graphs is that the strength of connect-edness between any pair of nodes in a graph is either 0 or 1 where as in fuzzy graphs, it can be any real number in [0,1].Consequently paths and cycles also have this property. Also as far as the applications are concerned (information networks,electric circuits, etc.), the reduction of flow between pairs of nodes is more relevant and may frequently occur than the totaldisruption of the flow or the disconnection of the entire network.

In view of this, in order to study such properties of information networks, Rosenfeld [9] defined the concept of a fuzzycutnode as a node in a fuzzy graph whose removal reduces the strength of connectedness between some pair of nodes. Sim-ilarly a fuzzy bridge is an arc whose removal from the fuzzy graph reduces the strength of connectedness between some pairof nodes. Also in 1975, Yeh and Bang [21] introduced two connectivity parameters in fuzzy graphs namely vertex connec-tivity and edge connectivity and used them in fuzzy graph clustering.

This paper is motivated mainly by the article of Yeh and Bang [21]. No further study was carried out on the parametersdefined in this article due to their limitations. Vertex connectivity ðXðGÞÞ [21] is defined as minimum weight of a disconnec-tion in G. A disconnection of a fuzzy graph is a vertex set D whose removal results in a disconnected or single vertex fuzzygraph. Edge connectivity ðkðGÞÞ of a fuzzy graph is defined to be the minimum weight of cut sets of G and a cut-set is the setof arcs between some bipartition of the vertex set of G. Clearly these definitions of vertex connectivity and edge connectivityare close to graphs rather than to fuzzy graphs since a disconnection of the fuzzy graph is essential to study about theseparameters. Thus the concepts of disconnection and cut-set [21] do not generalize fuzzy cutnodes and fuzzy bridges inthe sense of Rosenfeld [9]. Thus it is natural that fuzzy node cuts and fuzzy arc cuts must be sets that reduce strength ofconnectedness between some pair of nodes in the fuzzy graph and the same has been explored in this paper.

Also, in modern fuzzy graph theory, we have the notions of strong as well as strongest paths [4] between any pair of nodesand a fuzzy arc cut (similarly for fuzzy node cut) can be viewed as a set of strong arcs whose removal from G reduces thestrength of connectedness between some pair of nodes of G, at least one of them differing from the end nodes of arcs inthe cut. The arcs which are not strong need not be considered because the flow through such arcs can be redirected througha different path having more strength. Also since both parameters of Yeh and Bang are related to the disconnection of thefuzzy graph, the ‘fuzziness’ in the situation cannot be studied properly. This is the motivation for introducing new connec-tivity concepts, fuzzy node connectivity ðjÞ and fuzzy arc connectivity ðj0Þ in a fuzzy graph. Also, this paper presents rela-tionship between the new and old connectivity parameters and generalizes one of the clustering techniques in [21].

Consider a fuzzy graph G : ðr;lÞ on four nodes a; b; c and d with rðaÞ ¼ rðbÞ ¼ rðcÞ ¼ rðdÞ ¼ 1 and lða; bÞ ¼ 0:8;lðb; cÞ ¼ 0:6;lðc; dÞ ¼ 0:4;lðd; aÞ ¼ 0:6;lðd; bÞ ¼ 0:35. A disconnection in G with minimum weight is D ¼ fb; dg and itsweight is 0.7. Hence XðGÞ ¼ 0:7. Also a cut set in G with minimum weight is given by E ¼ fðd; cÞ; ðc; bÞg with weight 1. HencekðGÞ ¼ 1. But this fuzzy graph is a fuzzy tree and it contains 3 fuzzy bridges and 2 fuzzy cutnodes. Thus we can find out sets ofnodes and arcs, reducing the strength of connectedness between some pair of nodes rather than a disconnection of G. In G,{a} and {b} are fuzzy node cuts (Definition 6) with minimum strength 0.6 each and hence j ¼ 0:6. Also, {a,d} and {b,c} arefuzzy arc cuts (Definition 11) with minimum strength 0.6 each and hence j0 ¼ 0:6. Thus without a disconnection, thestrength of connectedness has been reduced in G.

Both distance based [9] and connectivity based [21] clustering techniques can be found in the literature. Yeh and Bangintroduced various types of clustering techniques like single linkage, k-linkage, k-edge connectivity, k-vertex connectivityand complete linkage processes. Also there are many other clustering algorithms in the theory like min-cut tree method,cohesiveness matrix method [21] etc. and they are all based on the parameters defined by Yeh and Bang. If one apply thenew definition of fuzzy arc cut and the proposed connectivity parameters, which is more fuzzy, it is possible to find moreclusters. Also, when the variables are of qualitative type, the new method can produce better results.

Section 2 contains preliminaries and in Section 3, the concept of strong degree in fuzzy graphs (Definition 1) and its prop-erties (Propositions 1–3) are discussed. Also in this section it is shown that a node with minimum node strength in a com-plete fuzzy graph(CFG) will have minimum strong degree and a node with maximum node strength will have maximumstrong degree (Lemma 1). The minimum and maximum strong degrees in a CFG are evaluated (Lemma 1). The concept ofnode strength sequence is introduced (Definition 3) in Section 4. The concepts of fuzzy node cut (Definition 6), fuzzy nodeconnectivity (Definition 9), fuzzy arc cut (Definition 12) and fuzzy arc connectivity (Definition 16) are introduced in Sections5 and 6. Fuzzy bonds (Definition 13) are introduced in Section 6. It is observed that at least one of the end nodes of a fuzzybond is a fuzzy cutnode and not conversely (Proposition 6). Here it is also shown that the fuzzy node connectivity and fuzzy

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arc connectivity of a fuzzy tree are equal and is equal to the minimum strength of a strong arc in it (Theorem 1). The fuzzyanalogue of Whitney’s theorem in graph theory which gives the relation between node connectivity, arc connectivity andminimum degree in a graph is obtained (Theorem 2) in Section 7. In a CFG, fuzzy node connectivity and fuzzy arc connec-tivity are equal and is equal to the minimum strong degree (Corollary 1). In Section 8, the relations between the connectivityparameters k and X in [21] and the proposed parameters, fuzzy node connectivity ðjÞ and fuzzy arc connectivity ðj0Þ arestudied (Theorems 5, 6). Also it is shown that all these parameters coincide in a CFG and is equal to the minimum strongdegree in it (Theorem 7). Finally in Section 9, the advantage of the new parameters over the existing parameters in clusteranalysis is mentioned.

2. Preliminaries

A fuzzy graph (f-graph) [7] is a pair G : ðr;lÞ where r is a fuzzy subset of a set S and l is a fuzzy relation on r. We assumethat S is finite and nonempty, l is reflexive and symmetric [7]. In all the examples r is chosen suitably. Also, we denote theunderlying graph by G� : ðr�;l�Þ where r� ¼ fu�S : rðuÞ > 0g and l� ¼ fðu;vÞ�S� S : lðu;vÞ > 0g. A fuzzy graph G : ðr;lÞ iscalled trivial if G� is trivial.

A path P of length n is a sequence of distinct nodes u0;u1; . . . ;un such that lðui�1;uiÞ > 0; i ¼ 1;2; . . . ;n and the degree ofmembership of a weakest arc is defined as its strength. The strength of connectedness between two nodes x and y is defined asthe maximum of the strengths of all paths between x and y and is denoted by CONNGðx; yÞ [9]. An x—y path P is called a stron-gest x—y path if its strength equals CONNGðx; yÞ [9]. An f-graph G : ðr;lÞ is connected if for every x; y in r�;CONNGðx; yÞ > 0.Through out, we assume that G is connected. An arc of an f-graph is called strong [4] if its weight is at least as great as theconnectedness of its end nodes when it is deleted. If ðx; yÞ is a strong arc, then x and y are strong neighbors. A node z is calleda fuzzy endnode (f-endnode) if it has exactly one strong neighbor in G [5]. An x—y path P is called a strong path if P containsonly strong arcs [4].

An arc is called a fuzzy bridge (f-bridge) of G if its removal reduces the strength of connectedness between some pair ofnodes in G [9]. Similarly a fuzzy cutnode (f-cutnode) w is a node in G whose removal from G reduces the strength of connect-edness between some pair of nodes other than w [9]. A fuzzy graph H : ðs; mÞ is called a partial fuzzy subgraph of G : ðr;lÞ ifsðuÞ 6 rðuÞ for every node u and mðu;vÞ 6 lðu;vÞ for every u;v [7]. In particular, we call H : ðs; mÞ, a fuzzy subgraph ofG : ðr;lÞ if sðuÞ ¼ rðuÞ for every u�s� and mðu;vÞ ¼ lðu;vÞ for every ðu;vÞ�m� [7]. A fuzzy sub graph H : ðs; mÞ spans the fuzzygraph G : ðr;lÞ if s ¼ r. The fuzzy graph H : ðs; mÞ is called a fuzzy subgraph of G induced by P if P � r�; sðuÞ ¼ rðuÞ for all u inP and mðu;vÞ ¼ lðu;vÞ for every u;v�P. A connected f-graph G : ðr;lÞ is a fuzzy tree (f-tree) if it has a fuzzy spanning subgraphF : ðr; mÞ, which is a tree, where for all arcs ðx; yÞ not in F there exists a path from x to y in F whose strength is more thanlðx; yÞ [9]. Note that F is the unique maximum spanning tree (MST) of G [15]. A complete fuzzy graph (CFG) is an f-graphG : ðr;lÞ such that lðx; yÞ ¼ rðxÞ ^ rðyÞ for all x and y [3].

The degree of a node v is defined as dðvÞ ¼P

u–vlðu;vÞ. The minimum degree of G is dðGÞ ¼ ^fdðvÞjv 2 r�g and the max-imum degree of G is DðGÞ ¼ _fdðvÞjv 2 r�g [8,21].

3. Strong degree of a node

Bhutani and Rosenfeld [4] introduced the concepts of strong arcs and strong paths. These concepts motivated researchersto reformulate some of the concepts in fuzzy graph theory more effectively. Bhutani and Rosenfeld have shown the existenceof a strong path between any two nodes in an f-graph. Sameena and Sunitha [13] have defined the strong degree of a node inan f-graph. The concept of strong degree is relevant in fuzzy graph applications especially problems related with flows as theflow through arcs, which are not strong, can be redirected through a different strongest path.

We summarize some of the results in [13] below.

Definition 1 [13]. Let G : ðr;lÞ be a fuzzy graph. The strong degree of a node v 2 r� is defined as the sum of membershipvalues of all strong arcs incident at v. It is denoted by dsðvÞ. Also if NsðvÞ denote the set of all strong neighbors of v, thendsðvÞ ¼

Pu2NsðvÞlðu;vÞ.

Example 1. Let G : ðr;lÞ be an f-graph with r� ¼ fu;v ;wg and lðu;vÞ ¼ 0:2;lðv ;wÞ ¼ 0:1 and lðw;uÞ ¼ 0:3. HeredsðuÞ ¼ 0:5; dsðvÞ ¼ 0:2; dsðwÞ ¼ 0:3.

Bhutani and Rosenfeld [4] have shown the existence of a strong path between any two nodes of a fuzzy graph and hencewe can find at least one strong arc incident at each node of a nontrivial connected f-graph and hence we have the followingproposition.

Proposition 1. In a non trivial connected f-graph G : ðr;lÞ;0 < dsðvÞ 6 dðvÞ for all nodes v 2 r�.

As in graphs, we can define the minimum and maximum strong degree of a node in an f-graph as given below.

Definition 2 [13]. The minimum strong degree of G is dsðGÞ ¼ ^fdsðvÞjv 2 r�g and maximum strong degree of G isDsðGÞ ¼ _fdsðvÞ;v 2 r�g.

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Example 2. Let G : ðr;lÞ be with r� ¼ fu;v;w; xg and lðu;vÞ ¼ 0:1 ¼ lðx;uÞ;lðv ;wÞ ¼ 1 ¼ lðu;wÞ;lðw; xÞ ¼ 0:3. In G, allarcs except ðu;vÞ and ðu; xÞ are strong. Thus dsðuÞ ¼ 1 ¼ dsðvÞ; dsðwÞ ¼ 2:3 and dsðxÞ ¼ 0:3. Hence dsðGÞ ¼ 0:3 and DsðGÞ ¼ 2:3.

Remark 1. As in the case of graphs, dsðGÞ 6 dsðvÞ 6 DsðGÞ;8v 2 r�. Also dðvÞ ¼ dsðvÞ for every node v in a graph.

Next we have an obvious result.

Proposition 2 [13]. The sum of strong degrees of all nodes in an f-graph is equal to twice the sum of membership values of allstrong arcs in G.

By proposition 3.13 of [3] and by definition of strong arcs [4], all arcs of a CFG are strong and hence dsðvÞ ¼ dðvÞ for allv 2 r�. Also strong degree of a node v in a CFG is given by dsðvÞ ¼

Pu–v ^ frðuÞ;rðvÞg where u 2 r� [13].

Proposition 3 [13]. In a CFG there exists at least one pair of nodes u and v such that dsðuÞ ¼ dsðvÞ.

Next lemma is related to the minimum and maximum degrees of a CFG. Since lðu;vÞ ¼ rðuÞ ^ rðvÞ for all arcs ðu;vÞ of aCFG G, the minimum and maximum degrees of nodes in G can be evaluated in terms of the membership values of its nodes.

Lemma 1. Let G : ðr;lÞ be a CFG with r� ¼ fu1;u2; . . . ;ung such that rðu1Þ 6 rðu2Þ 6 rðu3Þ 6 � � � 6 rðunÞ. Then ðu1;ujÞ is an arcof minimum weight at uj for 2 6 j 6 n and ðui;unÞ is an arc of maximum weight at ui for 1 6 i 6 n� 1. Also,

dðu1Þ ¼ dsðGÞ ¼ ðn� 1Þrðu1Þ and

dðunÞ ¼ DsðGÞ ¼Xn�1

i¼1

rðuiÞ:

Proof. Throughout the proof, we suppose that rðu1Þ < rðu2Þ 6 rðu3Þ 6 � � � 6 rðun�1Þ < rðunÞ. If there are more than onenode with minimum node strength or maximum node strength, the proof will be similar. First we prove that for2 6 j 6 n; ðu1; ujÞ is an arc of minimum weight at uj. If possible, suppose that ðu1;ulÞ;2 6 l 6 n is not an arc of minimumweight at ul. Also let ðuk;ulÞ;2 6 k 6 n; k–l be an arc of minimum weight at ul. Being a CFG,

lðu1;ulÞ ¼ rðu1Þ ^ rðulÞ and,lðuk;ulÞ ¼ rðukÞ ^ rðulÞ.Since lðuk; ulÞ < lðu1;ulÞ, we have,rðukÞ ^ rðulÞ < rðu1Þ ^ rðulÞ ¼ rðu1Þ.

That is either rðukÞ < rðu1Þ or rðulÞ < rðu1Þ. Since l; k – 1, this is a contradiction to our assumption that rðu1Þ is theunique minimum node degree. Thus for 2 6 j 6 n; ðu1;ujÞ is an arc of minimum weight at uj.

Next, we prove that ðui;unÞ is an arc of maximum weight at ui for 1 6 i 6 n� 1. On the contrary suppose thatðuk;unÞ;1 6 k 6 n� 1 is not an arc of maximum weight at uk and let ðuk;urÞ;1 6 r 6 n� 1; k – r be an arc of maximumweight at uk.

Then, lðuk;urÞ > lðuk;unÞ and hence,rðukÞ ^ rðurÞ > rðukÞ ^ rðunÞ ¼ rðukÞ, which implies that rðurÞ > rðukÞ:Therefore, lðuk;urÞ ¼ rðukÞ ¼ lðuk; unÞ, which is a contradiction to our assumption. Thus ðuk;unÞ is an arc of maximum

weight at uk.Now we have,

dsðu1Þ ¼Xn

i¼2

lðu1;uiÞ ¼Xn

i¼2

ðrðu1Þ ^ rðuiÞÞ ¼Xn

i¼2

rðu1Þ ¼ ðn� 1Þrðu1Þ:

If possible suppose that dsðu1Þ – dsðGÞ and let uk; k – 1 be a node in G with minimum strong degree.Now dsðu1Þ > dsðukÞ; implies

Xn

i¼2

lðu1; uiÞ >X

k–1;j–k

lðuk;ujÞ:

That is

Xn

i¼2

ðrðu1Þ ^ rðuiÞÞ >X

k–1;j–k

ðrðukÞ ^ rðujÞÞ:

Since rðu1Þ ^ rðuiÞ ¼ rðu1Þ for i ¼ 2;3; . . . ;n;rðukÞ ^ rðu1Þ ¼ rðu1Þ and for all other indices j;rðukÞ ^ rðujÞ > rðu1Þ, hence itfollows that

ðn� 1Þrðu1Þ >X

k–1;j–k

ðrðukÞ ^ rðujÞÞ > ðn� 1Þrðu1Þ:

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That is, dsðu1Þ > dsðu1Þ, a contradiction.Thus dsðu1Þ ¼ dsðGÞ ¼ ðn� 1Þrðu1Þ.Finally, we show that dsðunÞ ¼ DsðGÞ ¼

Pn�1i¼1 rðuiÞ.

Since rðunÞ > rðuiÞ for i ¼ 1;2; . . . ;n� 1 and G is a CFG,lðun;uiÞ ¼ rðunÞ ^ rðuiÞ ¼ rðuiÞ:

Therefore, dsðunÞ ¼Pn�1

i¼1 lðun; uiÞ ¼Pn�1

i¼1 rðuiÞ.Now if possible suppose that dsðunÞ – DsðGÞ. Let ul;1 6 l 6 n� 1 be a node in G such that dsðulÞ ¼ DsðGÞ and

dsðunÞ < dsðulÞ.Now,

dsðulÞ ¼Xl�1

i¼1

lðui;ulÞ þXn�1

i¼lþ1

lðui;ulÞ þ lðun; ulÞ:

6

Xl�1

i¼1

rðuiÞ þ ðn� lÞrðulÞ þ rðulÞ 6Xn�1

i¼1

rðuiÞ ¼ dsðunÞ:

That is, dsðulÞ 6 dsðunÞ, a contradiction to our assumption. Thus the Proposition is proved. h

4. Node strength sequence

In graphs, all nodes are assumed to have the same membership value 1, whereas in fuzzy graphs the membership value ofeach node is always a real number in (0,1]. So to each fuzzy graph, we can associate a sequence of real numbers namely thenode strength sequence (n–s sequence) as given below.

Definition 3. Let G : ðr;lÞ be an f-graph with jr�j ¼ n. Then the node-strength sequence (n-s sequence) of G is defined to beðp1; p2; . . . ; pnÞ with p1 6 p2 6 � � � 6 pn where pi;0 < pi 6 1 is the strength of node i when nodes are arranged so that theirstrengths are non decreasing. In particular p1 is the smallest node strength and pn, the largest node strength.

The following example illustrates this concept.

Example 3. Let G : ðr;lÞ be with r� ¼ fa; b; c; dg and rðaÞ ¼ rðcÞ ¼ rðdÞ ¼ 0:1;rðbÞ ¼ 0:2. Then the node-strength sequenceof G is (0.1,0.1,0.1,0.2) or ð0:13;0:2Þ:

By observing the n–s sequence, one can determine the number of nodes of minimum strong degree and maximum strongdegree in a CFG as in the next proposition.

Proposition 5. Let G : ðr;lÞ be a CFG with jr�j ¼ n. Then,(i) If the n–s sequence of G is of the form pn�1

1 ; p2

� �, then dsðGÞ ¼ DsðGÞ ¼ ðn� 1Þp1 ¼ dsðuiÞ; i ¼ 1;2; . . . ;n.

(ii) If the n–s sequence of G is of the form pr11 ; p

n�r12

� �with 0 < r1 6 n� 2, then there exist exactly r1 nodes with degree dsðGÞ and

n� r1 nodes with degree DsðGÞ.(iii) If the n–s sequence of G is of the form pr1

1 ; pr22 ; . . . ; prk

k

� �with rk > 1 and k > 2, then there exist exactly r1 nodes with degree

dsðGÞ and exactly rk nodes with degree DsðGÞ.(iv) If the n–s sequence of G is of the form pr1

1 ; pr22 ; . . . ; prk�1

k�1 ; pk

� �with k > 2, then there exist exactly 1þ rk�1 nodes with degreeDsðGÞ.

Proof. The proofs of (i) and (ii) are obvious. We present the proofs for (iii) and (iv).(iii) Let v ðjÞi ; j ¼ 1;2; . . . ; ri be the set of nodes in G with ds v ðjÞi

� �¼ pi;1 6 i 6 k. By Lemma 1, we have,

ds v ðjÞ1

� �¼ dsðGÞ ¼ ðn� 1Þp1 for j ¼ 1;2; . . . ; r1:

No node with strength more than p1 can have degree dsðGÞ since,l v ðjÞi ;v

ðlÞiþ1

� �¼ r v ðjÞi

� �> p1 for 2 6 i 6 k; j ¼ 1;2; . . . ; ri; l ¼ 1;2; . . . ; riþ1. Thus there exists exactly r1 nodes with strong

degree dsðGÞ.Next we prove that dsðv t

kÞ ¼ DsðGÞ; t ¼ 1;2; . . . ; rk.Since r v t

k

� �is the maximum node strength, we have l v t

k;vjk

� �¼ pk; t – j; t; j ¼ 1;2; . . . ; rk and l v t

k;vji

� �¼

r v tk

� �^ r v j

i

� �¼ r v j

i

� �for t ¼ 1;2; . . . ; rk; j ¼ 1;2; . . . ; ri; i ¼ 1;2; . . . ; k� 1.

Thus for t ¼ 1;2; . . . ; rk,

ds v tk

� �¼Xk�1

i¼1

Xri

j¼1

r v ji

� �þ ðrk � 1Þpk:

¼Xn�1

i¼1

rðuiÞ ¼ DsðGÞ: ðBy Lemma 1Þ

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Now if u is a node such that rðuÞ ¼ pk�1, we have,

dsðuÞ ¼Xk�2

i¼1

Xri

j¼1

lðu;v jiÞ þ ðrk�1 � 1þ rkÞpk�1:

¼Xk�2

i¼1

Xri

j¼1

r v ji

� �þXrk�1

j¼1

r v jk�1

� �þ ðrk � 1Þpk�1:

<Xk�2

i¼1

Xri

j¼1

r v ji

� �þXrk�1

j¼1

r v jk�1

� �þ ðrk � 1Þpk ¼ DsðGÞ:

Thus there exists exactly rk nodes with degree DsðGÞ.(iv) Let v ð1Þk ¼ vk be the node in G such that dsðvkÞ ¼ pk.

Then by Lemma 1, dsðvkÞ ¼ DsðGÞ ¼Pn�1

i¼1 rðuiÞ.Now let v t

k�1; t ¼ 1;2; . . . ; rk�1 be the nodes in G with ds v tk�1

� �¼ pk�1.

Then for t ¼ 1;2; . . . ; rk�1,

dsðv tk�1Þ ¼

Xk�2

i¼1

Xri

j¼1

l v ji; v

tk�1

� �þXl–m

l v lk�1;v

mk�1

� �þ l v l

k�1;vk

� �:

But, l v ji;v t

k�1

� �¼ r v j

i

� �for i ¼ 1;2; . . . ; k� 2 and j ¼ 1;2; . . . ; ri.

l v lk�1;v

mk�1

� �¼ pk�1 and l v l

k�1;vk� �

¼ pk�1:

Thus, ds v tk�1

� �¼Pk�2

i¼1

Prij¼1r v j

i

� �þ ðrk�1 � 1Þpk�1 þ pk�1.

¼Xk�2

i¼1

Xri

j¼1

r v ji

� �þ rk�1pk�1:

¼Xn�1

i¼1

rðuiÞ ¼ DsðGÞ:

Thus there exist rk�1 þ 1 nodes with strong degree DsðGÞ.Now if u is a node such that rðuÞ < pk�1, as in the proof of (iii), we can show that dsðuÞ < DsðGÞ. Thus there exist exactly

rk�1 þ 1 nodes with strong degree DsðGÞ and the proof is complete. h

5. Fuzzy node connectivity

Yeh and Bang [21] introduced two connectivity parameters of a fuzzy graph namely vertex connectivity and edge connec-tivity in their paper titled ‘‘Fuzzy relations, Fuzzy graphs and their applications to clustering analysis”. In this section we gener-alize these definitions using the concepts of strong arcs. As mentioned in the introduction, both node connectivity and arcconnectivity [21] are related with sets disconnecting the fuzzy graph. But in fuzzy set up, we need only the reduction ofstrength of connectedness between some pair of nodes. The definitions of disconnection and vertex connectivity of Yeh andBang is given below.

Definition 4 [21]. A disconnection of a fuzzy graph G : ðr;lÞ is a vertex set D whose removal results in a disconnected or asingle vertex graph. The weight of D is defined to be

Pv2Dfminlðv ;uÞjlðv ;uÞ – 0g.

Definition 5. The vertex connectivity of a fuzzy graph G, denoted by XðGÞ, is defined to be the minimum weight of a discon-nection in G.

We generalize these definitions as follows.

Definition 6. Let G : ðr;lÞ be a connected f-graph. A set of nodes X ¼ fv1;v2; . . . ;vmg � r� is said to be a fuzzy node cut(FNC) if either, CONNG�Xðx; yÞ < CONNGðx; yÞ for some pair of nodes x; y 2 r� such that both x; y – v i for i ¼ 1;2; . . . ;m orG� X is trivial.

If there are n nodes in X, then X is called an n-FNC. Clearly a 1-FNC is a singleton set X ¼ fug where u is an f-cutnode.

Example 4. Let G : ðr;lÞ be with r� ¼ fa; b; c; dg and lða; bÞ ¼ lðc; dÞ ¼ 0:3;lða; dÞ ¼ lðb; cÞ ¼ 0:2;lða; cÞ ¼ 0:1. ThenS ¼ fb; dg is a 2-FNC for, CONNG�Sða; cÞ ¼ 0:1 < 0:2 ¼ CONNGða; cÞ.

Example 5. Let G : ðr;lÞ be with r� ¼ fa; b; cgwith lða; bÞ ¼ lðc; aÞ ¼ 0:1;lðb; cÞ ¼ 0:2. G has no f-cutnodes but all the threepairs of nodes are fuzzy node cuts since the removal of any pair of nodes results in a trivial f-graph.

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In [4], it is shown that there exist at least one strong arc incident on every vertex of a nontrivial connected fuzzy graph.Motivated by this, we have the following definition.

Definition 7. Let X be a fuzzy node cut in G. The strong weight of X, denoted by sðXÞ is defined as sðXÞ ¼P

x2Xlðx; yÞ, wherelðx; yÞ is the minimum of the weights of strong arcs incident on x.

Definition 8. The fuzzy node connectivity of a connected fuzzy graph G is defined as the minimum strong weight of fuzzynode cuts of G. It is denoted by jðGÞ.

Example 6. Let G : ðr;lÞ be with r� ¼ fa; b; c; dg and lða; bÞ ¼ 0:1;lðb; cÞ ¼ 0:4;lðc; dÞ ¼ 0:3;lðd; aÞ ¼ lða; cÞ ¼ 0:2. ThenX1 ¼ fcg is the only 1-FNC (i.e., c is a fuzzy cutnode) with sðX1Þ ¼ 0:2. The only 2-FNC in G is X2 ¼ fa; cg and sðX2Þ ¼ 0:4. Alsoany three nodes of G form a 3-FNC with sðfa; b; cgÞ ¼ sðfa; b; dgÞ ¼ sðfb; c; dgÞ ¼ 0:8 and sðfa; c; dgÞ ¼ 0:6. Thus jðGÞ ¼ 0:2.

6. Fuzzy arc connectivity

In [21], the notion of arc connectivity of an f-graph is defined as given below. As mentioned in the introduction this def-inition is more close to a graph rather than a fuzzy graph since, in a fuzzy graph the concept of strength of connectednessplays a crucial role.

The following definitions of cut-set and edge connectivity are due to Yeh and Bang [21].

Definition 9. Let G be a fuzzy graph and fV1;V2g be a partition of its vertex set. The set of edges joining vertices of V1 andvertices of V2 is called a cut-set of G, denoted by ðV1;V2Þ relative to the partition fV1;V2g. The weight of the cut-set ðV1;V2Þ isdefined as

Pu2V1 ;v2V2

lðu;vÞ.

Definition 10. Let G be a fuzzy graph. The edge connectivity of G denoted by kðGÞ is defined to be the minimum weight ofcut-sets of G.

Now we give a more general definition of fuzzy arc cuts as follows. If the removal of a set of strong arcs E from a givenfuzzy graph reduces the strength of connectedness not only between the end nodes of arcs in E, but also between some pairof nodes at least one of them differing from the end nodes of arcs in E, we call it a fuzzy arc cut. The arcs which are not strongneed not be considered because such arcs do not affect the strength of connectedness between any pair of nodes.

Definition 11. Let G : ðr;lÞ be an f-graph. A set of strong arcs E ¼ fe1; e2; . . . :engwith ei ¼ ðui;v iÞ; i ¼ 1;2; . . . ;n is said to be afuzzy arc cut (FAC) if either CONNG�Eðx; yÞ < CONNGðx; yÞ for some pair of nodes x; y 2 r� with at least one of x or y is differentfrom both ui and v i; i ¼ 1;2; . . . ;n, or G� E is disconnected.

If there are n arcs in E then it is called an n-FAC.Among all fuzzy arc cuts, an arc cut with one arc (1-FAC) is a special type of fuzzy bridge and we have,

Definition 12. A 1-FAC is called a fuzzy bond (f-bond).

Remark 2. Note that f-bonds are special type of f-bridges. Not all f-bridges are f-bonds. For example, f-bridges of an f-tree aref-bonds (Theorem 9 of [16]). Note that all bridges in a graph are bonds.

Example 7. Let G : ðr;lÞ be with r� ¼ fa; b; c; d; eg with lða; bÞ ¼ 0:3;lða; cÞ ¼ 0:1;lðb; cÞ ¼ 0:4;lðc; dÞ ¼ 0:5;lðd; aÞ ¼ 0:6;lða; eÞ ¼ lðb; eÞ ¼ 0:5. There are 4 fuzzy bonds(1-FAC) in this fuzzy graph namely arcs ða; dÞ; ða; eÞ; ðd; cÞ and ðe; bÞ. AlsoE ¼ fða; bÞ; ðd; cÞg is a 2-FAC since 0:4 ¼ CONNG�Eðe; cÞ < CONNGðe; cÞ ¼ 0:5.

As noted in Remark 2, all the f-bridges of an f-tree are f-bonds. But there are other examples of non f-trees with this prop-erty as seen from the following example.

Example 8. Let G : ðr;lÞ be with r� ¼ fa; b; c; d; g with lða; bÞ ¼ 1;lðb; cÞ ¼ 0:5;lðc; dÞ ¼ 0:1;lðd; aÞ ¼ 0:2;lða; cÞ ¼ 0:3;lðb; dÞ ¼ 0:2. Here G is not an f-tree and there are two f-bridges namely arc ða; bÞ and arc ðb; cÞ which are f-bonds becausedeletion of each of these arcs from G reduces the strength of connectedness between a and c from 0.5 to 0.3.

In graphs, if ðu; vÞ is a bridge, then at least one of u or v must be a cutnode. But in fuzzy graphs, if ðu;vÞ is a fuzzy bridge, itis not necessary that at least one of u or v is a fuzzy cutnode. Note that blocks in fuzzy graphs and CFG can contain fuzzybridges but no fuzzy cunodes [9,16]. But in an f-bond, we have,

Proposition 6. At least one of the end nodes of an f-bond is an f-cutnode.

Proof. Let G : ðr;lÞ be an f-graph and e ¼ ðu;vÞ be an f-bond in G. Being an f-bond, the deletion of e from G reduces thestrength of connectedness between x and y with at least one of them is different from u and v. If both x and y are differentfrom u and v, then u as well as v will be f-cutnodes. If one of x or y coincides with u or v, then u or v which is neither x nor ywill be an f-cutnode. h

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Remark 3. Since blocks and complete fuzzy graphs contain no f-cutnodes, from Proposition 6, it follows that no f-bridges ina CFG or in a block are f-bonds. Also, in an f-tree, if an f-bond has only one of its end nodes as an f-cutnode, then the other endnode will be an f-end node.

Next we define the strong weight of a FAC and the fuzzy arc connectivity of a fuzzy graph.

Definition 13. The strong weight of a fuzzy arc cut E is defined as s0ðEÞ ¼P

ei2ElðeiÞ.

Definition 14. The fuzzy arc connectivity j0ðGÞ of a connected fuzzy graph G is defined as the minimum strong weight offuzzy arc cuts of G.

Example 9. Let G : ðr;lÞ be with r� ¼ fa; b; c; dg with lða; bÞ ¼ 0:1;lðb; cÞ ¼ 0:8;lðc; dÞ ¼ 0:7;lðd; aÞ ¼ 0:3;lða; cÞ ¼ 0:3.Then E1 ¼ fðb; cÞg and E2 ¼ fðc; dÞg are the only 1-FACs (fuzzy bonds) of G with s0ðE1Þ ¼ 0:8 and s0ðE2Þ ¼ 0:7. ButE3 ¼ fða; dÞ; ða; cÞg is a 2-FAC in G with weight s0ðE3Þ ¼ 0:6: Among all fuzzy arc cuts of G, E3 has the minimum strong weightand hence j0ðGÞ ¼ 0:6. Also, note that jðGÞ ¼ 0:3:

7. Fuzzy analogue of Whitney’s theorem

In a tree with at least three nodes, jðGÞ ¼ j0ðGÞ ¼ 1. This is due to the fact that all arcs in a tree are strong with strengthone and so we have the fuzzy analogue,

Theorem 1. In an f-tree G : ðr;lÞ, jðGÞ ¼ j0ðGÞ ¼ ^flðx; yÞ : ðx; yÞ is a strong arc in Gg.

Proof. Let G : ðr;lÞ be an f-tree. Consider the unique maximum spanning tree F of G. An arc ðx; yÞ in G : ðr;lÞ is an f-bridge ifand only if ðx; yÞ is an arc of the maximum spanning tree F : ðr; mÞ of G [9]. All these f-bridges are f-bonds by Remark 2.Also all arcs in F are strong. Thus each strong arc in F is a 1-FAC of G. Clearly the strong weight of each such 1-FAC islðx; yÞ. Hence fuzzy arc connectivity j0ðGÞ of G is the minimum weight of all arcs in F and hence the minimum weight ofall strong arcs in G.

Now every internal node of F is an f-cutnode of G [16] and hence are 1-fuzzy node cuts of G. Hence fuzzy nodeconnectivity jðGÞ of G is the minimum weight of all arcs in F and hence the minimum weight of all strong arcs in G. Hence theproof is complete. h

Remark 4. In a general fuzzy graph, Theorem 1 does not hold as seen from the following example.

Example 10. Let G : ðr;lÞ be with r� ¼ fa; b; c; dg with lða; bÞ ¼ 1;lðb; cÞ ¼ 0:6;lðc; dÞ ¼ 0:1;lðd; aÞ ¼ 0:25;lða; cÞ ¼ 0:2;lðb; dÞ ¼ 0:25. Here G is not an f-tree. Arcs ða; bÞ and ðb; cÞ are the only f-bridges in G. Clearly these are f-bonds(1-FACs).Now, s0ðfða; bÞgÞ ¼ 1 and s0ðfðb; cÞgÞ ¼ 0:6. But the fuzzy arc connectivity of G is 0.5 as E ¼ fðd; aÞ; ðd; bÞg is a 2-FAC withminimum strong weight. Note that removal of E from G reduces the strength of connectedness between d and c from0.25 to 0.1.

Next we present the fuzzy analogue of a famous result regarding node connectivity, edge connectivity and minimum de-gree of a graph due to Hassler Whitney.

Theorem 2. In a connected f-graph G : ðr;lÞ;jðGÞ 6 j0ðGÞ 6 dsðGÞ:

Proof. First we shall prove the second inequality. Let G : ðr;lÞ be a connected f-graph. Let v be a node in G such thatdsðvÞ ¼ dsðGÞ. Let E be the set of strong arcs incident at v. If these are the only arcs incident at v, then G� E is disconnected.If not, let ðv ;uÞ be an arc which is not strong incident at v. Then u is a node different from the end nodes of arcs in E. Bydefinition of a strong arc,

lðu;vÞ < CONNGðu; vÞ;

which implies that there exists a strongest u� v path say P in G which should definitely pass through one of the strong arcsat v. Thus the removal of E from G will reduce the strength of connectedness between v and u. Thus in both cases, E is a fuzzyarc cut. The strong weight of this FAC is dsðGÞ. Hence, it follows that j0ðGÞ 6 dsðGÞ.

Next to prove jðGÞ 6 j0ðGÞ, let E be a FAC with strong weight j0. We have the following cases.

Case-1: Every arc in E has one node in common v (say).In this case E ¼ fei ¼ ðv ;v iÞ; i ¼ 1;2; . . . ;ng.Let X ¼ fv1;v2; . . . ;vng. Then clearly X is a fuzzy node cut. Now,

minu2r�lðv i;uÞ 6 lðv ;v iÞ:

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Therefore,X

i

ðminu2r�lðv i;uÞÞ 6 lðv1;vÞ þ lðv2; vÞ þ � � � þ lðvn; vÞ:

That is jðGÞ 6 j0ðGÞ.Case-2: Not all arcs in E have a node in common.Let E ¼ fei ¼ ðui;v iÞ; i ¼ 1;2; . . . ; ng for some n.Let X1 ¼ fu1;u2; . . . ;ung and X2 ¼ fv1;v2; . . . ;vng. By assumption, CONNG�Eðx; yÞ < CONNGðx; yÞ for some pair of nodesx; y 2 r� with at least one of x or y is different from both ui and v i for i ¼ 1;2; . . . ;n.Sub Case I: x and y are not members of X1

SX2.

In this case, take X ¼ X1 or X ¼ X2. Then clearly X is a fuzzy node cut since its deletion from G reduces the strength ofconnectedness between x and y and,

jðGÞ 6 strong weight of X 6 strong weight of E ¼ j0ðGÞ:Sub Case II: Either x or y is in X1

SX2.

Without loss of generality suppose that x is in X1S

X2. Let x 2 X1. Then take X ¼ X2. Clearly X is a fuzzy node cut, for; thedeletion of X from G will reduce the strength of connectedness between x and y. Thus,

jðGÞ 6 strong weight of X 6 strong weight of E ¼ j0ðGÞ:

Thus in all cases, jðGÞ 6 j0ðGÞ 6 dsðGÞ. h

The newly defined parameters coincide on a CFG and their values are equal to the minimum strong degree of the fuzzygraph as given in the next corollary.

Corollary 1. In a CFG, G : ðr;lÞ, jðGÞ ¼ j0ðGÞ ¼ dsðGÞ:

Proof. Let G : ðr;lÞ be a CFG such that jr�j ¼ n. Since G is complete, the deletion of any set E of n� 2 arcs from G will notreduce the strength of connectedness between any pair of nodes in G different from the nodes adjacent to the arcs in E (Thedeletion of a fuzzy bridge in G will reduce the strength of connectedness of its end nodes only [16]). Any set of n� 1 arcsincident at a node u in G is a FAC with strong weight dsðuÞ ¼ dðuÞ. Let v be a node in G such that dsðvÞ ¼ dsðGÞ. Clearly theset of arcs incident at v is a FAC with minimum strong weight.

Therefore j0ðGÞ ¼ dsðvÞ ¼ dsðGÞ.Now we prove that jðGÞ ¼ dsðGÞ.If possible suppose that jðGÞ – dsðGÞ. By Theorem 2, jðGÞ 6 j0ðGÞ 6 dsðGÞ. Hence jðGÞ < dsðGÞ.But since the deletion of i nodes 1 6 i 6 n� 2 results again in a nontrivial CFG, any FNC should contain at least n� 1

nodes. Among such fuzzy node cuts, the one which does not contain a node v such that dsðvÞ ¼ dsðGÞ say S1 will have theminimum strong weight since the set of arcs adjacent with nodes in S1 with one of its end at v are the arcs with minimumweights at nodes of S1.

Thus jðGÞ ¼ sðS1Þ < dsðGÞ.Now let E1 be the set of all arcs incident with the node v, then E1 is a FAC such that

s0ðE1Þ ¼ sðS1Þ < dsðGÞ:

Which contradicts the fact that j0ðGÞ ¼ dsðGÞ.Hence jðGÞ ¼ j0ðGÞ ¼ dsðGÞ. h

Remark 5. Even if the values of these parameters coincide on a fuzzy graph, it is not necessary that the given fuzzy graph is aCFG as seen from the following example.

Example 11. Let G : ðr;lÞ be an f-graph with r� ¼ fa; b; cg with rðaÞ ¼ 0:9;rðbÞ ¼ 1;rðcÞ ¼ 0:8;lða; bÞ ¼ 0:2;lðb; cÞ ¼ 0:1;lða; cÞ ¼ 0:1. Then jðGÞ ¼ j0ðGÞ ¼ dsðGÞ ¼ 0:2, but G is not a CFG.

8. Other connectivity parameters

In this section, we present relations between the new connectivity parameters and the existing Yeh and Bang parameters[21]. The new parameters always produce smaller values than the values of the existing parameters as seen from Theorem 5.

Theorem 3 [21]. Let G be an f-graph, then XðGÞ 6 kðGÞ 6 dðGÞ.

For given real numbers a, b and c, there exists a fuzzy graph with vertex connectivity a, edge connectivity b and the min-imum degree c.

Theorem 4 [21]. For any real numbers a, b and c such that 0 < a 6 b 6 c, there exist an f-graph G with XðGÞ ¼ a; kðGÞ ¼ b anddðGÞ ¼ c.

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Theorem 5. Let G : ðr;lÞ be a connected f-graph. Then j0ðGÞ 6 kðGÞ.

Proof. Let G : ðr;lÞ be a connected f-graph with edge connectivity kðGÞ. Let E ¼ ðV1;V2Þ be a cut-set in G with minimumweight. That is weight of E ¼ kðGÞ. Since E partitions the node set into two sets namely V1 and V2, the removal of E from Gdisconnects G. Let G1 : ðr1;l1Þ and G2 : ðr2;l2Þ be the fuzzy subgraphs of G induced by V1 and V2, respectively. Let x 2 r�1and y 2 r�2. Then, CONNG�Eðx; yÞ ¼ 0 < CONNGðx; yÞ. Hence E is a FAC in G. Now j0 being the minimum strong weight of allFACs, it follows that j0ðGÞ 6 weightðEÞ ¼ kðGÞ, which completes the proof. h

In an f-tree, edge connectivity and minimum degree are upper bounds for both jðGÞ and j0ðGÞ as seen from the followingtheorem.

Theorem 6. Let G : ðr;lÞ be an f-tree. Then jðGÞ ¼ j0ðGÞ 6 kðGÞ 6 dðGÞ:

Proof. Proof follows from Theorems 1, 3 and 5. h

Note that when the fuzzy graph is complete, all these four parameters coincide with value equal to minimum strong de-gree of the fuzzy graph. Thus we have,

Theorem 7. Let G : ðr;lÞ be a CFG. Then, jðGÞ ¼ j0ðGÞ ¼ XðGÞ ¼ kðGÞ ¼ dsðGÞ:

Proof. First we prove that in a CFG, kðGÞ ¼ dsðGÞ. Let G : ðr;lÞ be a CFG with jr�j ¼ n. Since G is complete the deletion of n� 2arcs will not disconnect the graph. So any cut-set in G will contain at least n� 1 arcs. Let v be a node in G such thatdsðvÞ ¼ dsðGÞ. Then since ðu; vÞ;u 2 r�; u – v is an arc with minimum weight at u, the cut-set ðV1;V2Þ with V1 ¼ fvg andV2 ¼ r� � fvg will have the minimum weight and by Lemma 1 it is equal to dsðvÞ ¼ dsðGÞ. So kðGÞ ¼ dsðGÞ; . . . ; ðiÞ.

Next we prove that XðGÞ ¼ dsðGÞ. Since G is complete, the deletion of i nodes, 1 6 i 6 n� 2 results again in a CFG.Therefore, any disconnection D will contain at least n� 1 nodes and the removal of D will results in a trivial fuzzy graph.Among such disconnections D, by Lemma 1, the one, not containing the node v will have the minimum weight.

Thus, XðGÞ ¼ dsðGÞ; . . . ; ðiiÞBy Corollary 1, jðGÞ ¼ j0ðGÞ ¼ dsðGÞ; . . . ; ðiiiÞNow from (i) (ii) and (iii)jðGÞ ¼ j0ðGÞ ¼ XðGÞ ¼ kðGÞ ¼ dsðGÞ. h

Remark 6. The condition in Theorem 7 is not sufficient for a fuzzy graph to be a CFG as seen from Example 11 above. G is nota CFG even if jðGÞ ¼ j0ðGÞ ¼ XðGÞ ¼ kðGÞ ¼ dsðGÞ ¼ 0:2.

9. Application to fuzzy graph clustering

The objective of clustering is to classify the observations into groups such that the degree of ‘association’ is high among themembers of a group and is less among the members of different groups. Graph theoretically clustering is nothing but partition-ing the graph based on qualitative aspects of the data. The first two articles on fuzzy graph theory by Rosenfeld [9] and Yeh andBang [21] were intended to present clustering techniques. Rosenfeld introduced distance based clustering while Yeh and Bangintroduced connectivity based clustering. Yeh and Bang presented a series of processes like single linkage, k-linkage, k-edgeconnectivity, k-vertex connectivity and complete linkage to extract fuzzy graph clusters. This paper is our main motivationand we modify the k-edge connectivity procedure using the newly defined parameters of connectivity. This modification isrequired due to the fact that the parameter ‘edge connectivity’ defined by Yeh and Bang is inclined more towards graphs thanfuzzy graphs. Using the new parameters of connectivity, we modify the old techniques and extract more clusters in a fuzzygraph. This modified procedure is more powerful than any of the methods discussed in [21] as seen from the illustration below.

In [21] Yeh and Bang has defined s – edge connected graphs and s – edge components of a fuzzy graph. If kðGÞ denotes theedge connectivity of a fuzzy graph G, then G is called s – edge connected if G is connected and kðGÞP s and a s – edge com-ponent of G is a maximal s – edge connected subgraph of G. Analogues to this using the concept of fuzzy arc connectivity j0,we have the following definitions.

Definition 15. A fuzzy graph G : ðr;lÞ is called t-fuzzy edge connected if G is connected and j0ðGÞP t for some t 2 ð0;1Þ:

Thus if j0ðGÞ ¼ t0, then G is t-fuzzy edge connected for all t such that t 6 t0:

Definition 16. A t-fuzzy edge component of G : ðr;lÞ is a maximal t-fuzzy edge connected fuzzy subgraph of G : ðr;lÞ.

Note that by maximal t-fuzzy edge connected subgraph, we mean a fuzzy subgraph H of G, induced by a set of nodes in Gsuch that j0ðHÞ ¼ t. The above concepts are illustrated in the following example.

Example 12. Let G : ðr;lÞ be an f-graph with r� ¼ fa; b; c; dg with rðaÞ ¼ rðbÞ ¼ rðcÞ ¼ rðdÞ ¼ 1 and lða; bÞ ¼ lða; cÞ ¼ 0:2;lðb; cÞ ¼ 0:3;lðb; dÞ ¼ 0:1;lðc; dÞ ¼ 0:4.

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Here j0ðGÞ ¼ 0:3. Hence G is t-fuzzy edge connected for all t such that t 6 0:3. Thus G itself is a t-fuzzy edge component for allt such that 0 < t 6 0:3. Next let t ¼ 0:4. Then the 0.4-fuzzy edge components of G are H1 ¼ hfa; b; cgi and H2 ¼ hfdgi. Herej0ðH1Þ ¼ 0:4.

Now using the definition of t-fuzzy edge components, we have the definition of a fuzzy cluster of level t as follows.

Definition 17. Let G : ðr;lÞ be an f-graph. A collection C of nodes in G is called a fuzzy cluster of level t if the fuzzy subgraphof G induced by C is a t-fuzzy edge component of G.

We use cohesive matrix [21] M to find the maximal t-edge connected components of a fuzzy graph G.

Definition 18 [21]. Let G : ðr;lÞ be a fuzzy graph. An element of G is defined to be either a node or an arc. The cohesivenessof an element denoted by hðeÞ, is the maximum value of edge connectivity of the subgraphs of G containing e.

Definition 19 [21]. Let G : ðr;lÞ be a fuzzy graph. The cohesive matrix M of G is defined as M ¼ ðmi;jÞwhere mi;j ¼ the cohe-siveness of the edge ðv i;v jÞ if i – j and the cohesiveness of the vertex v i if i ¼ j.

Note that a node v 2 r� is said to be in a cluster of level t if v belongs to a t-fuzzy edge component of G. Thus finding the t-fuzzy edge components of G is equivalent to the extraction of clusters from G. This process of finding t-fuzzy edge componentsand thus finding the fuzzy clusters in G based on fuzzy arc connectivity j0 is termed t-fuzzy edge connectivity procedure.

9.1. t-Fuzzy edge connectivity procedure

Step-1: Obtain the Cohesive matrix [21] M of the fuzzy graph G : ðr;lÞ.Step-2: Obtain the t-threshold graph Gt of M.Step-3: The maximal complete subgraphs of Gt are the t-fuzzy edge components.

9.2. Illustration: Cancer detection problem

Based on the location of the cells in the low magnification image of a tissue sample, surgically removed from a humanpatient, it is possible to construct a graph G with nodes as cells, called cell graph [22]. By analyzing the physical featuresof the cells; for example color and size, we can assign a membership value to the nodes of G. This value will range over(0,1] depending on the nature of the cell; that is healthy, inflammatory or cancerous. Also, arcs of G can assign a membershipvalue based on the distance between the cells. Thus the cell graph can be converted to a fuzzy graph in this manner. Applyingthe above clustering procedure to such a fuzzy graph, the cancerous cell clusters can be detected at the cellular level in prin-ciple. This process, classifies cell clusters in a tissue into different phases of cancer, depending on the distribution, densityand the fuzzy connectivity of the cell clusters within the tissue. Moreover, this process helps in examining the dynamicsand progress of the cancer qualitatively [22].

Consider the fuzzy graph G given by the following fuzzy matrix representing a fuzzy cell graph consisting of ten cells.Assume that the nodes with weights more than 0.5 represent cancerous cells, nodes with weights between 0.2 and 0.5inflammatory cells and between 0 and 0.2, healthy cells. Let the nodes of G be fa; b; c; d; e; f ; g;h; i; jg and let,

A ¼

0:00 0:13 0:00 0:00 0:00 0:10 0:00 0:00 0:00 0:000:13 0:00 0:30 0:00 0:00 0:00 0:15 0:00 0:00 0:000:00 0:30 0:00 0:50 0:00 0:00 0:00 0:40 0:00 0:000:00 0:00 0:50 0:00 0:90 0:00 0:00 0:00 0:70 0:000:00 0:00 0:00 0:90 0:00 0:00 0:00 0:00 0:00 1:000:10 0:00 0:00 0:00 0:00 0:00 0:14 0:00 0:00 0:000:00 0:15 0:00 0:00 0:00 0:14 0:00 0:20 0:00 0:000:00 0:00 0:40 0:00 0:00 0:00 0:20 0:00 0:60 0:000:00 0:00 0:00 0:70 0:00 0:00 0:00 0:60 0:00 0:800:00 0:00 0:00 0:00 1:00 0:00 0:00 0:00 0:80 0:00

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

:

The cohesive matrix M of G is given below.

M ¼

0:00 0:13 0:13 0:13 0:13 0:13 0:13 0:13 0:13 0:130:13 0:00 0:30 0:30 0:30 0:14 0:20 0:30 0:30 0:300:13 0:30 0:00 0:50 0:50 0:14 0:20 0:50 0:50 0:500:13 0:30 0:50 0:00 0:90 0:14 0:20 0:60 0:80 0:900:13 0:30 0:50 0:90 0:00 0:14 0:20 0:60 0:80 1:000:13 0:14 0:14 0:14 0:14 0:00 0:14 0:14 0:14 0:140:13 0:20 0:20 0:20 0:20 0:14 0:00 0:20 0:20 0:200:13 0:30 0:50 0:60 0:60 0:14 0:20 0:00 0:60 0:600:13 0:30 0:50 0:80 0:80 0:14 0:20 0:60 0:00 0:800:13 0:30 0:50 0:90 1:00 0:14 0:20 0:60 0:80 0:00

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

:

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Author's personal copy

For any value t 2 ð0;1Þ, we can find the threshold graph Gt from M. The t-fuzzy edge components are the maximal completesubgraphs of Gt . The corresponding nodes in these components form clusters of level t. For example, the threshold graph fort ¼ 0:5 is given below.

G0:5 ¼

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 1 1 0 0 1 1 10 0 1 0 1 0 0 1 1 10 0 1 1 0 0 0 1 1 10 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 1 1 1 0 0 0 1 10 0 1 1 1 0 0 1 0 10 0 1 1 1 0 0 1 1 0

0BBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCA

:

The different fuzzy clusters of level 0:5 obtained from G0:5 are fc; d; e;h; i; jg; fag; fbg; ffg and fgg.The fuzzy clusters of all levels are given below.

Level Fuzzy clustersð1;1Þ fag; fbg; fcg; fdg; feg; ffg; fgg; fhg; fig; fjgð0:9; 1� fe; jg; fag; fbg; fcg; fdg; ffg; fgg; fhg; figð0:8; 0:9� fd; e; jg; fag; fbg; fcg; ffg; fgg; fhg; figð0:6; 0:8� fd; e; i; jg; fag; fbg; fcg; ffg; fgg; fhgð0:5; 0:6� fd; e; h; i; jg; fag; fbg; fcg; ffg; fggð0:3; 0:5� fc; d; e; h; i; jg; fag; fbg; ffg; fggð0:2; 0:3� fb; c; d; e; h; i; jg; fag; ffg; fggð0:14; 0:2� fb; c; d; e; g; h; i; jg; fag; ffgð0:13; 0:14� fb; c; d; e; f ; g; h; i; jg; fagð0; 0:13� fa; b; c; d; e; f ; g; h; i; jg

From the above fuzzy clusters corresponding to t ¼ 0:5 (which is the threshold for cancerous cells), it is observed thatfd; e;h; i; jg is a cell cluster which is affected seriously by cancer whereas its neighboring area containing the cells b, c andg can be found inflammatory. Note that the cells a and f are healthy.

9.3. Comparison between new and existing methods

As mentioned above, t – fuzzy edge connectivity procedure is more powerful than the s – edge connectivity procedure of Yehand Bang as can be seen from the following example.

Consider the fuzzy graph in Example 12. The clusters using s – edge connectivity procedure [21] and fuzzy clusters using t –fuzzy edge connectivity procedure are as follows.

9.4. s – Edge connectivity procedure: [21]

The edge connectivity of the f-graph G in Example 12 is k ¼ 0:4.Using the Yeh and Bang procedure, we obtain the s – edge components of G as given below.

Level Maximal s – e.c subgraphs Clusters

(0,0.4] hfa; b; c; dgi C1 ¼ fa; b; c; dg(0.4,1] hfagi; hfbgi; hfcgi; hfdgi C2 ¼ fag;C3 ¼ fbg;C4 ¼ fcg;C5 ¼ fdg

By this, we get only two types of clusters corresponding to all possible levels namely the full set of nodes and the clustersof singletons. We will find more clusters if we apply the t – fuzzy edge connectivity procedure as seen below.

9.5. t – fuzzy edge connectivity procedure

The fuzzy arc connectivity j0 of the f-graph G in Example 12 is 0.3. The clusters of level t are obtained as follows.

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Level Maximal t – f.e.c. subgraphs Fuzzy clusters

(0,0.3] hfa; b; c;dgi C1 ¼ fa; b; c; dg(0.3,0.4] hfa; b; cgi; hfdgi C2 ¼ fa; b; c; g;C3 ¼ fdg(0.4,1] hfagi; hfbgi; hfcgi; hfdgi C4 ¼ fag;C5 ¼ fbg;C6 ¼ fcg;C7 ¼ fdg

Thus we get three types of clusters corresponding to different levels. When we deal with qualitative data, this will def-initely produce more clusters. If the parameter in the above procedure represents the degree of interaction among four re-search students, then, by the existing method we may observe that there is minimum interaction between all the studentsand there are no groups of high interaction. But using the proposed method we can find a group with more positive inter-action and can identify that there is a student who is less active in the whole group.

10. Concluding remarks

Connectivity concepts are the key in graph clustering and networks. The connectivity parameters in fuzzy graphs intro-duced by Yeh and Bang were not properly studied there after. There are several applications for the parameters of Yeh andBang. But they are not ‘fuzzy’ parameters as they deal with disconnections of fuzzy graphs. This motivated us to reformulatethese concepts in a ‘strong’ sense.

We have introduced two connectivity parameters in fuzzy graphs. The newly defined parameters and the existing onesare compared and obtained the relations between them. A generalization to the Whitney’s theorem is obtained.

The old connectivity parameters in fuzzy graphs are directly applied in clustering problems. The new parameters also canbe similarly and effectively applied in clustering. We have applied fuzzy arc connectivity in cluster extraction. The remainingmethods will be considered in the forthcoming papers.

Acknowledgements

We thank Professor Witold Pedrycz, Chief Editor, Information Sciences and the reviewers for their valuable suggestions inimproving the quality of this paper.

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